Escape regimes of biased random walks on Galton–Watson trees

Abstract

We study biased random walk on subcritical and supercritical GaltonWatson trees conditioned to survive in the transient, sub-ballistic regime. By considering offspring laws with infinite variance, we extend previously known results for the walk on the supercritical tree and observe new trapping phenomena for the walk on the subcritical tree which, in this case, always yield sub-ballisticity. This is contrary to the walk on the supercritical tree which always has some ballistic phase.

Biased Random Walk on the Trace of Biased Random Walk on the Trace of …

“…This is a consequence of the law of large numbers for X (0) and the law of large numbers for regeneration times that is stated at Lemma 2.1(c). In conjunction with [2, Theorem 2.11], (7) yields that Z (i) is a p (i) -recurrent graph.…”

“…Moreover, in the latter cases, due to a lattice effect that is also seen in other models, it might be anticipated that the relevant scaling limits only hold along subsequences. See [7] for recent progress in the Galton-Watson tree case, and [4] for a discussion of results and conjectures in the percolation case.…”

“…As early as the 1980s, physicists observed that such phenomenon might be relevant when the random graphs are percolation clusters, empirically demonstrating the nonmonotonicity of the speed, and sub-ballisticity in the strong bias regime [3]. Mathematically, a phase transition between ballisticity and sub-ballisticity was first shown rigorously for the simpler model of random walk on supercritical Galton-Watson trees [14] (see also [5,7,9] for recent work concerning more detailed properties of such processes), and has since been confirmed to hold in the percolation setting [6,13,19]. A relatively up-to-date survey of these developments is given in [4].…”

Central limit theorems for biased randomly trapped random walks onZ

“…The ballistic phase for this walk is studied further in [1] where an expression of the speed is given and in [4] where appropriate scaling sequences for the sub-ballistic phase are shown. The traps formed in the supercritical tree resemble those in the subcritical tree and it has been shown in [8] that the walks observe similar scaling regimes.…”

“…In this paper we are only concerned with ballistic walks. We note that the sub-ballistic regimes for the biased walk on the subcritical tree have been studied in [8] where it is shown that either a strong bias or heavy tails of the offspring law can slow the walk into a sub-ballistic phase.…”